Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{10}{5+4 \sin \theta} $$

Short answer

The graph is an ellipse centered vertically due to \( \sin \theta \) and the given parameters.

Step-by-step solution

Identify the Conic Type

The given polar equation is of the form \( r = \frac{ed}{1 + e \sin \theta} \). For our equation, \( e = 4 \), which is the coefficient of \( \sin \theta \), and \( d = 2.5 \) as expressed from the factor with 10 in the numerator divided by 4. This indicates that the conic is an ellipse because \( 0 < e < 1 \).

Determine the Orientation

Since the equation is \( r = \frac{10}{5 + 4 \sin \theta} \), it shows that the \( \sin \theta \) term is used, which indicates that the major axis of the ellipse is vertical, aligning with the y-axis in a polar grid.

Find the Directrix and Major Axis

For ellipses in polar coordinates of this form, noting that \( e = \frac{4}{5} \), the distance from the pole (origin) to the directrix is \( \frac{d}{e} = \frac{2.5}{4/5} = 3.125 \). This provides the vertical line or a value for plotting purposes.

Understand the Range of \( r \) and Graph

To sketch the graph, consider various angles \( \theta \). For \( \theta = 0° \), \( r = \frac{10}{5} = 2 \). For \( \theta = 90° \) or \( 270° \), \( \sin \theta = \pm 1 \) making \( r = \frac{10}{1} = 10 \) and making \( r \) values maximum and minimum at these orientations. Continue to plot these values around a polar coordinate grid, ensuring the correctness in proportions.

Sketch the Graph

Utilizing the information about how \( r \) changes with angle based on the prior step, sketch the general shape of an ellipse centered around the vertical line through the origin, ensuring its consistency with calculated points and the general limits outlined.

Key concepts

Ellipses

Ellipses are one of the fundamental shapes that belong to the family of conic sections. An ellipse is formed by slicing a cone with a plane at an angle, which does not intersect the base of the cone. This results in a symmetrical, closed curve that resembles an elongated circle. In mathematics, ellipses are defined by a specific set of points where the sum of the distances from two fixed points, known as the foci, remains constant.

In polar coordinates, ellipses can be expressed in the form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \), depending on their orientation. Here, \( e \) is the eccentricity, which describes how stretched the ellipse is (\( 0 < e < 1 \) for ellipses). \( d \) represents the distance from the focus to the directrix, a fixed line used in the definition of conics.

The properties of an ellipse make it important in physics and engineering, as it often describes orbits of celestial bodies and pathways of projectiles.

Polar Coordinates

Polar coordinates provide a way of describing a point in the plane using an angle and a distance from a reference point, rather than using a pair of coordinates \((x, y)\) on a Cartesian grid. In this system, each point is determined by its distance \( r \) from the origin, called the pole, and an angle \( \theta \), measured from a fixed direction.

One of the main advantages of polar coordinates is that they simplify the representation of curves, particularly those that are symmetrical around a point, such as circles or spirals. Polar coordinates are particularly useful in scenarios where the relationship between points is angularly centered around a single point.

• The radius \( r \) can be positive or negative; a negative value indicates the point is in the opposite direction to the angle \( \theta \).
• The angle \( \theta \) can be given in degrees or radians, and it acts as the rotational measure from the positive x-axis.

Understanding polar coordinates is crucial for graphing complex shapes like ellipses, as it provides a clearer representation when the equations involve trigonometric components like \( \sin \) or \( \cos \).

Graph Sketching

Graph sketching is a key skill in mathematics that involves drawing the basic shape or outline of a mathematical function or relation. It helps visualize the behavior and properties of equations, particularly useful in the case of conic sections such as ellipses.

When sketching graphs from polar equations, consider these steps:

• Identify the type of conic section. In our example, with the equation \( r = \frac{10}{5 + 4 \sin \theta} \), it is identified as an ellipse.
• Find the orientation—whether the major axis is vertical or horizontal. For ellipses in polar form, observe the trigonometric component used (\( \sin \theta \) or \( \cos \theta \)).
• Determine constant values such as the eccentricity \( e \), directrix, and focal points, as these help locate the shape on the graph.
• Calculate specific values of \( r \) for strategic angles like \( 0^\circ \), \( 90^\circ \), and sometimes \( 270^\circ \), to locate key points.

Finally, sketch the outline, ensuring to capture the symmetry and relative proportions indicated by the polar equation. This process, breaking down the graph into manageable components, results in an accurate representation of complex shapes.